Advice on upgrading an old dual-system HVAC

I've bought the house six months ago. Built in 1955 with forced air gas furnace heating 3-bed/1-bath single story, roughly 825 sq ft. Furnace was last replaced in 2009, and is 108,000 BTU. No central A/C. I have three window A/C units in this part of the house.

House was renovated in 1976, single-room and half-bath addition (275sq ft) built onto main floor, and 3-bed/3-bath basement (1100 sq ft) finished. Basement and main floor addition are heated by gas boiler/baseboard radiators, original 1976 still in use, 91,500 BTUs. There is another window AC unit down there, plus two through-the-wall AC units downstairs and one in the upstairs addition.

The boiler is obviously going to go any day now at its age. It's become pretty noisy and the pressure when I moved in was WAY too high (red zone). I'm looking to replace it, and depending on the solution I go with maybe add a cooling system as well. I'm meeting with a couple HVAC companies next week but wanted to have a good understanding of my options before talking to them so I can be a smart consumer.

My home inspector last summer suggested I look into ductless mini-splits when I replaced the boiler. I'm a little confused about how these work when used for heating AND cooling - so much of the online info is cooling-focused.

* Do mini-splits have to be powered by a heat pump, or can they be hooked up to a furnace or boiler somehow?

* I read that mini-splits actually can be used with ducts even though they're designed not to need them. Is there an advantage to doing that with pre-existing ductwork? I had the ducts heavily insulated and the house is well-sealed so energy loss in both ducts and mini-splits should be minimal.

* I have a decently large house (around 2100 sq ft total, 6-bed/4.5 bath, currently using 7 window/wall AC units and would likely need that many mini-split units) and it wasn't clear to me if I could get a system powerful enough for all that, especially without paying an arm and a leg. The estimates I've found for installing a HE gas boiler are around $7500 and it looked like it might cost that much just to install two mini-split units which seems crazy! I know that pricing is not discussed here but some sense of the order of magnitude would be helpful - ie is it closer to $7500 or closer to $35000.

* Would it be cheaper to do some sort of combination system? Could a single pump supply heat and cooling through the ducts in that part of the house and through mini-splits in the renovation? Or would I still need to keep the furnace to heat the original house and have the pump just heat the renovation? If either setup is an option, would it be cheaper to keep the furnace and do a smaller mini-split system, given the furnace will be more expensive to operate?

* Are there other options I'm overlooking? I'd prefer not to put in new radiators unless budget gives me little other choice, as I don't like the look and find them hard to keep free of dust and grime. (I'm aware of the downsides of forced air as well, w/r/t dry air and allergens &c, but would rather deal with that than dusty, grimy, ugly radiators). How involved/expensive of a job is putting in new radiant floor heating?

* Environmental concerns are important to me and I'm willing to pay a little bit more for something with lower emissions, even if that doesn't translate into lower operating costs. I am still probably looking at a total budget of no more than $10,000.